Weekly series of agricultural prices usually exhibit seasonal variations and the stationarity of these variations should be taken into
account to analyse price relationships. However, unit root tests at seasonal frequencies are unlikely to have good power properties.
Furthermore, movements in actual price series are often not as expected when unit roots are present. Therefore, stationarity tests
at seasonal frequencies also need to be applied. In this paper, a procedure to test for the null hypothesis of stationarity at seasonal
frequencies was extended to the weekly case. Once critical values were obtained by simulation exercises, unit root and stationarity tests
were applied to weekly retail prices of di?erent agricultural commodities in Spain. The most relevant fnding was that many unit roots
that seasonal unit root tests failed to reject did not seem to be present from the results of seasonal stationarity tests, whereas seasonal
unit root tests led to the rejection of some unit roots that seemed to be present according to the results of seasonal stationarity tests.
In conclusion, unit root tests should be complemented with stationarity tests before making decisions about the behaviour of seasonal
patterns.

Additional keywords: agricultural prices; weekly series; unit roots.

Authors' contributions: Both authors participated in the conception and design of the research, methodological proposal, analysis
of data and writing of the paper.

In research on agricultural prices, seasonal effects in a season are usually assumed to be fixed over the sample period. Therefore, such effects are modelled by means of seasonal dummies. However, as commented by Cáceres-Hernández & Martín-Rodríguez (2017), wrong assumptions about the seasonal component may lead to erroneous conclusions about the dynamic behaviour of the series and the transmission mechanisms between them1. Moreover, as explained by Meyer & Von-Cramon-Taubadel (2004), data frequency plays a crucial role in attempts to identify these important effects to assess agricultural and commercial policies. Therefore, weekly series, increasingly available, could be needed to quantify some dynamic relationships between prices.

1Seasonal unit roots force long run relationships and error correction models to be reformulated. As indicated by Palaskas & Crowe (1996), when the
presence of seasonal unit roots is ignored, unit root and cointegration tests are found to lack consistency and power. However, the application of inadequate flters to remove potential seasonal roots is a bad solution, due to the distortions in the estimates of the dynamic process of transmission effects
between prices.

The testing procedures described in the following sub-section were applied to weekly series of retail agricultural prices (in €/kg) from 2006 to 20163. Original data are openly available from the web page of the Spanish Ministry of Agriculture, Food and Environment4. To avoid weekly series with a high number of missing values, the products finally chosen were: 9 types of vegetables (chards, courgettes, onions, lettuces, beans, potatoes, peppers, tomatoes, carrots), 4 types of fruits (apples, bananas, lemons, pears), eggs, 5 types of meat (pork, rabbit, lamb, chicken and veal) and 12 types of fish (anchovy, blue whiting, mackerel, baby clam, john dory, horse mackerel, mussel, hake, small hake, salmon, sardine, trout)5.

Test for seasonal unit roots and stationarity

In the first part of this sub-section, the procedure proposed by Cáceres-Hernández (1996) for testing the null hypothesis of seasonal unit roots is described. In the second part, the auxiliary regressions and the statistics for testing the null hypothesis of stationarity are explained.

a) Test for seasonal unit roots

Let the data generating process for the weekly series {yt}t=1,…,T be given by

where f(B) is an autoregressive polynomial, dtrepresents the deterministic component (trend plus seasonal), and et is a white noise disturbance term. The length of the seasonal period is assumed to be 52 weeks.

To test for seasonal unit roots in weekly series, the procedure described in Cáceres-Hernández (1996), following Franses (1991), can be applied. The following auxiliary regression needs to be estimated,

where Δ52 (B)=1-B52, and regressors y1,t,…,y27,t are defined as

A number of lags of the dependent variable are included in order to ensure serial uncorrelation in the error term. Then, the hypothesis of unit root at zero frequency is rejected when the null hypothesis p1=0 is rejected against p1<0 by means of a t type test t1. The hypothesis of unit root at Nyquist frequency is rejected when the null hypothesis p2=0 is rejected against p2<0 by means of another t type test t2. As regards the remainder of seasonal frequencies, an F type test Fk-2about the significance of parameters pk,1, pk,2, can be applied to test the presence of a pair of unit roots at the seasonal frequency θk,k=3,…,276. In this paper, critical values to these tests are obtained by means of simulation exercises adapted to the sample size of the series analysed.

b) Test for stationarity

To test for the null hypothesis of stationarity at zero and seasonal frequencies, the procedure described in Khedhiri & Montasser (2012), following Kwiatkowski et al. (1992), can be applied. Let the data generating process for the series {yt}t=1,…,T be again given by Eq. (1). Once the regressors y1,t,…,y27,t are defined as in Eqs. (3.a) to (3.c) to isolate the effects of other unit roots in the series, the following auxiliary regressions need to be estimated.

The test for the presence of unit root at zero frequency is obtained by assuming that the data generating process for the series {y1,t}t=1,…,T is such that

where

and ut and vt are zero mean weakly dependent dis­turbance terms. Then, by estimating the following auxiliary regression

the statistic similar to the one proposed in Kwia­tkowski et al. (1992) is calculated as

where

and

The test for the presence of unit root at the Nyquist frequency was obtained by assuming that the data generating process for the series {y2,t}t=1,…,T is such that:

If the original series {yt}t=1,…,T is assumed to be stationary around a deterministic component, the auxiliary regression for testing the null hypothesis of stationarity at any frequency is

This being the case, to test for the stationarity hypothesis at a frequency a filtering procedure to remove other unit roots was not necessary. Once this auxiliary regression was estimated, the statistical tests η(0), η(p) and η(θk), k=3,…,27, could be calculated from the residuals of such an estimation.

The asymptotic distribution of the test statistic η(0) is the one which was obtained in Kwiatkowski et al. (1992), whereas for statistics η(p) and η(θk), k=3,…,27, the corresponding asymptotic distributions were the same as those obtained by Khedhiri & Montasser (2012). Note that the asymptotic distribution of statistics η(θk), k=3,…,27, was the same at any frequency, and, as shown in Montasser (2015), the frequency of observation had not effect on the asymptotic distribution of statistics η(p) and η(θk). In this paper, critical values were obtained by simulation exercises adapted to the sample size for price series7.

4In the original source, there are 53 weekly observations corresponding to years 2009 and 2014. However, in order to obtain a fxed number of seasons,
the decision has been made to substitute observations corresponding to weeks 26 and 27 by an average of these two observations. Furthermore, missing
values at weeks 51 and 52 in 2006, at week 1 in 2007, and at week 6 in 2010 have been assigned an average of the corresponding contiguous observations.

5For the blue whiting series, an anomalous observation and other six missing data were substituted by an average of the corresponding contiguous observations.

6Del Barrio-Castro & Sansó (2015) showed that the distribution of the t-ratio unit root tests associated to the zero and Nyquist frequencies and also for
the F-type tests associated to the harmonic frequencies are asymptotically equivalent to the corresponding distributions obtained when the regressors
defned in Hylleberg et al. (1990) are applied.

7Te TSP fles to obtain critical values are included as supplementary material accompanying the paper on SJAR’s website.

The tests for zero and seasonal frequencies proposed in the previous section were applied to the weekly price series already mentioned. To assess the instability of the seasonal patterns in these series, a previous approach to these variations was obtained as the difference between original and 52-week moving average series8. Then, an evolving periodic cubic spline has been adjusted to these differences9. The results of estimating such splines are shown in Figures 1 and 2. According to these figures, seasonal patterns do not seem to be fixed, but these patterns do not change as much as expected when seasonal unit roots are present.

Following the conventional procedure to test for seasonal unit roots, a linear trend and seasonal dummies are included as deterministic components in the auxiliary regressions10. However, the slope term has been removed when it is statistically non significant. Furthermore, the results of residual autocorrelation tests show that lags of the dependent variable do not need to be included. Table 1 shows the critical values obtained for the effective sample size (572, 11 years of weekly data). Given that the sample distribution of seasonal unit root tests depends on the deterministic components in the data generating process, Monte Carlo simulation experiments have been designed to obtain critical values depending on the inclusion of a slope term in the auxiliary regression.

Tables 2 and 3 show the values of the statistics for testing the null hypothesis of unit root at zero and seasonal frequencies11. Besides the results for the zero frequency, which should be analysed once a conclusion is obtained with regard to seasonal frequencies, the unit root tests fail to reject the null hypothesis at some seasonal frequencies for some price series. At 10% significance level, the unit root hypothesis was not rejected for potato, lemon and pear prices at frequency p. At the same significance level, the tests also failed to reject the null hypothesis for lemon prices at frequencies p/26, 2p/26 and 6p/26, and for pear prices at frequency 5p/26. At 5% significance level, the null hypothesis was not rejected for onion prices at frequency 8p/26, for bean prices at frequency 25p/26, for pepper prices at frequency p, for apple prices at frequency 14p/26, for lemon prices at frequency 3p/26, and for pear prices at frequency 10p/26. In the case of egg prices, the unit root was rejected at frequency p at 5% significance level.

With regard to meat and fish price series, unit root tests failed to reject the null hypothesis at 10% significance level for hake prices at frequency 4p/26, for pork and sardine prices at frequency p, for rabbit prices at frequency 2p/26, and for salmon prices at frequencies 13p/26 and 16p/26. At 5% significance level, the unit root hypothesis was rejected for john dory prices at frequency 4p/26, for blue whiting prices at frequency 5p/26, for hake and salmon prices at frequency 15p/26, for sardine prices at frequency 2p/26, and for trout prices at frequency 18p/26.

Finally, Tables 4 and 5 show the results of testing the null hypothesis of stationarity at zero and seasonal frequencies by estimating the auxiliary regression in Eq. (24). In order for the non-parametric correction of the estimate of the error variance to take the serial correlation into account, the maximum length, l , is set at 3 or 8, following conventional criteria based on the sample size (Newey & West, 1987). Only the minimum values of the test statistics corresponding to these two values of parameter l are shown. According to the critical values in Table 1, and leaving aside the rejection of the null hypothesis at the zero frequency for most of the series, the stationarity hypothesis was rejected for bean prices at frequencies p/26 and 2p/26, for lamb prices at frequency p/26 and also for sardine prices at frequency 2p/26 at 5% significance level, whereas at 10% significance level the null hypothesis was rejected for lemon prices at frequency p/2612.

Note that many unit roots that seasonal unit root tests failed to reject did not seem to be present from the results of seasonal stationarity tests. Furthermore, seasonal unit root tests led to the rejection of some unit roots that seemed to be present according to the results of seasonal stationarity tests.

8To obtain estimates of seasonal e?ects in the frst half of 2006 and in the second half of 2016, moving average series at these points in time have been
calculated using prices observed in 2005 and 2017.

9A spline is a piecewise polynomial function which provides smooth estimates of seasonal effects and allow us to observe the changes in the shape of the
seasonal pattern. It has been selected a six-segment cubic spline as defned in Cáceres-Hernández & Martín-Rodríguez (2017) when restrictions between
years are not imposed. Tat is to say, spline parameters evolve from year to year whereas break points are located in fxed points for every year. Tese
positions are chosen to minimize the sum of squared residuals when such a spline is ftted to the difference series.

10Note that spline functions are not applied as a model for the deterministic seasonal component in auxiliary regressions.

11Te seasonal difference flter was applied to the original series from 2005 to 2016 in such a way that the effective sample size to estimate auxiliary regression was 572.

12Te rejection of the stationarity hypothesis may become non rejection when the original series were fltered of all unit roots except the one corresponding
to the frequency tested, but it is not clear these unit roots were present.

Seasonal patterns in agricultural price series usually exhibit changes such that unit root tests fail to reject the null hypothesis at some seasonal frequencies. However, these changes were not as variable as expected when these seasonal unit roots are causing them. In these circumstances, and taking the bad power performance of unit root tests into account, stationarity tests should also be applied as a complementary testing procedure. The conclusion regarding the presence of a unit root may be right when both procedures lead to such a conclusion. However, when doubts about the presence of seasonal unit roots remain after applying unit root and stationarity tests, some reflections are needed about the behaviour of the seasonal patterns. It should be noted that these testing procedures only take two possibilities into account (unit root or stationarity around a fixed deterministic component), but changes in the deterministic component of the seasonal pattern are another alternative to be explored, as pointed out by Cáceres-Hernández & Martín-Rodríguez (2017), before making a final decision about the presence of seasonal unit roots.

Likewise, a note of caution should be mentioned about the results of these testing procedures when applied to weekly series with small sample sizes. The number of observations corresponding to the same season is usually low in available agricultural price series. Therefore, the changes in the seasonal effect corresponding to a season are not easily observed. Of course, as commented by Hyndman & Kostenko (2007), the minimum sample size requirements increase with the amount of random variation in the data. Furthermore, economic knowledge about agricultural market performance is a key element to identify such changes in price behavior and, obviously, this knowledge is also very useful to model price relationships.

Afonso-Rodríguez JA, Santana-Gallego M, 2014. An analysis of the seasonal persistence for the time series of international monthly tourist arrivals in the Canary Islands. IX Seminario Canario de Economía, Empresa y Turismo, La Laguna (Spain), June 19-20.